Question

Please give only answer (b)

Answer 1

(x - 5)/(x³ + x² - 6x)
= (x - 5)/x(x² + x - 6)
=(x - 5)/x(x² + 3x -2x - 6)
= (x - 5)/x(x + 3)(x - 2)

Let (x - 5)/x(x + 3)(x - 2) = A/x + B/(x + 3) + C/(x - 2)
now, for finding value of A , ( assume B and C are not available )
I mean,
(x - 5)/x(x + 3)(x - 2) = A/x
multiply both sides with x .
(x - 5)/(x + 3)(x - 2) = A
now, put x = 0
(0 - 5)/(0 + 3)(0 - 2) = A
-5/-6 = A
A = 5/6

similarly, for finding value of B take similar steps .
B {x = -3 } = (x - 5)/x(x - 2)
B = {(-3) - 5}/(-3)(-3-2)
= { -8}/(-3)(-5) = -8/15
B = -8/15

similarly for C ,
C {x = 2} = (x - 5)/x(x + 3 )
C = ( 2 - 5)/2(2 + 3)
= -3/10
C = -3/10

hence, after resolved
$\frac{ {x} - 5 }{ {x}^{3} + {x}^{2} - 6x } = \frac{5}{6x} + \frac{-8}{15(x + 3)} + \frac{-3}{10(x - 2)}$